arXiv:astro-ph/0001167v1 11 Jan 2000 nXryflxwt rqece xedn 10 exceeding frequencies with flux qu and X-ray periodic in ph oscillations, fundamental millisecond of on measurements concentrate perform I to D them 30 use on to launch Ti tempts after X-Ray years hav 2.5 Rossi within expectations NASA’s 1993) early with al. discoveries These et 1 of Bradt Shara (RXTE; series 1978). 1982; a Joss in also al. verified see et dur Alpar 1982, surface 1984, Bath star’s Srinivasan the & on ca (Radhakrishnan form which patterns periods, emission spin millisecond asymmetric reach Accretin quasi-perio ac will 1973). cause stars an (Sunyaev neutron millisecond will in a stars about orbiting of neutron scales clumps time and date hot holes that black example, insight around For an object, (1971). compact Shvartsman proces stellar-mass the a gr in onto of dynam occur ter influence msec) naturally the (0.1–1 will under short variability matter Millisecond the o of expressions is motion phy basic objects the the most characterizing compact on the these window of of unique One ness a matter. provide dense objects and gravity these that is holes Introduction 1 Astrophysic and Astronomy 2000 of September Review appear Annual to the to Submitted in aes a ense eryecuieyfo low-magn from exclusively nearly seen been far propertie so spectral have X-ray tions and variability slower to lations nti eiw ics hs el icvrdpeoeaa phenomena discovered newly these discuss I review, this In st neutron accreting studying for motivation principal The measu perform to systems. them these use in t to interest physical attempts review ones mental the related I describe possibly and and matter. dates, phenomena dense neutron-star and and new star gravity these invo neutron strong the phenomena of around new closely understanding motion these orbital for and Models oscilla spin, burst contai oscillations. pulsations, binaries millisecond periodic Ex X-ray stars: Timing low-mass neutron X-ray from field Rossi observed the are with phenomena discovered accr from been phenomena recently variability have X-ray millisecond first The ILSCN SILTOSI -A BINARIES X-RAY IN OSCILLATIONS MILLISECOND srnmclIsiue“no anke” nvriyo A of University Pannekoek”, “Anton Institute Astronomical rila 0,19 JAsedm h Netherlands The Amsterdam, SJ 1098 403, Kruislaan -al [email protected] E-mail: .VNDRKLIS DER VAN M. 1 2 . 5 z u lodsusterre- their discuss also I but Hz, in n ioet quasi- kiloHertz and tions tn opc objects compact eting s-eidcvariations asi-periodic eyepiil nour on explicitly rely v h eto star neutron the lve faceino mat- of accretion of s .Mlieodoscilla- Millisecond s. nbakhl candi- black-hole in iglow-magnetic- ning eosrain of observations he lrr he new Three plorer. o-antcfield low-magnetic g edtce when detected be n eet ffunda- of rements akt tleast at to back s i aiblt on variability dic tcfil neutron etic-field n -a bursts X-ray ing vt erthem. near avity msterdam h compact- the f igExplorer ming cltm scale time ical sclinterest. ysical cme 1995. ecember nlybeen finally e r n black and ars iso strong of sics 8,Lvo& Livio 982, rto disk cretion dteat- the nd s; stars, so these will be the focus of this review, although I shall compare their phenomenology to that of the black-hole-candidates ( 6-7). § Accreting neutron stars and black holes occur in X-ray binaries (e.g., Lewin et al. 1995a). In these systems matter is transferred from a normal (’donor’) star to a compact object. Thermal X-rays powered by the gravitational po- tential energy released are emitted by the inner regions of the accretion flow and, if present, the neutron star surface. For a compact object with a size of order 101 km, 90% of the energy is released in the inner 102 km. It is with this inner emitting region that we shall be mostly concerned∼ here. Because accreting low-magnetic-field neutron stars are mostly found in low-mass X-ray binaries (in which the donor star has a mass of <1M⊙) these systems will be the ones we focus on. The mass transfer usually occurs by way of an accretion disk around the compact object. In the disk the matter moves in near-Keplerian orbits, i.e., with an azimuthal velocity that is approximately Keplerian and a radial veloc- ity much smaller than this. The disk has a radius of 105−7 km, depending on the binary separation. The geometry of the flow in the inner emitting regions is uncertain. In most models for accretion onto low-magnetic-field neutron stars (e.g., Miller et al. 1998a) at least part of the flow extends down into the emitting region in the form of a Keplerian disk. It is terminated either at the radius R of the star itself, or at a radius rin somewhat larger than R, by for example the interaction with a weak neutron-star magnetic field, radiation drag, or relativistic effects. Within rin the flow is no longer Keplerian and may or may not be disk-like. Both inside and outside rin matter may leave the disk and either flow in more radially or be expelled. Particularly for black holes advective flow solutions are discussed where the disk terminates and the flow becomes more spherical at a much larger radius (e.g., Narayan 1997). Whatever the geometry, it is clear that as the characteristic velocities near the compact object are of order (GM/R)1/2 0.5c, the dynamical time scale, the time scale for the motion of matter through∼ the emitting region, is short; τ (r3/GM)1/2 0.1 ms for r=10 km, and 2 ms for r=100km near dyn ≡ ∼ ∼ a 1.4M⊙ neutron star, and 1ms at 100km from a 10M⊙ black hole. So, the significance of millisecond∼ X-ray variability from X-ray binaries is clear: milliseconds is the natural time scale of the accretion process in the X-ray emitting regions, and hence strong X-ray variability on such time scales is nearly certainly caused by the motion of matter in these regions. Orbital motion, neutron-star spin, disk- and neutron-star oscillations are all expected to happen on these time scales. The inner flow is located in regions of spacetime where strong-field general- relativity is required to describe the motion of matter. For that reason one

2 expects to detect strong-field general-relativistic effects in these flows, such as for example the existence of a region where no stable orbits are possible. The precise interactions between the elementary particles in the interior of a neu- tron star which determine the equation of state (EOS) of supra-nuclear-density matter are not known. Therefore we can not confidently predict the radius of a neutron star of given mass, or the maximum spin rate or mass of neutron stars (e.g., Cook et al. 1994). So, by measuring these macroscopic quantities one constrains the EOS and tests basic ideas about the properties of elementary particles. In summary, the main motivation for studying millisecond variations in X-ray binaries is that their properties depend on untested, or even unknown, properties of spacetime and matter. Three different millisecond phenomena have now been observed in X-ray binaries. Historically, the first to be discovered were the twin kilohertz quasi- periodic oscillations (kHz QPOs), widely interpreted now as due to orbital motion in the inner accretion flow. Then came the burst oscillations, probably due to the spin of a layer in the neutron star’s atmosphere in near-corotation with the neutron star itself. Finally RXTE detected the first true spin fre- quency of an accreting low-magnetic field neutron star, the long-anticipated accreting millisecond pulsar. In this review, I first examine the millisecond pulsar ( 3), then the burst oscillations ( 4) and finally the kHz QPOs ( 5). We will thus§ be venturing from the (relatively)§ well-understood accreting pulsars§ via the less secure regions of what happens in detail on a neutron star’s surface during the thermonuclear runaway that is an X-ray burst, into the mostly uncharted territory of the innermost accretion flows around neutron stars and black holes, which obvi- ously is “where the monsters are”, but also where the greatest rewards wait. The possibly related phenomena found in black-hole candidates ( 6.1) and at lower frequencies ( 6.3) are discussed next, and in 7 the kHz QPO§ models are summarized. § §

2 Techniques

Most of the variability measurements discussed here rely on Fourier analysis of X-ray count-rate time series with sub-millisecond time resolution (van der Klis 1989b). A quasi-periodic oscillation (QPO) in the time series stands out in the power spectrum (the square of the Fourier transform) as a broad, usually Lorentzian peak (in Fig.5 several of such peaks can be seen), characterized by its frequency ν (“centroid frequency”), width λ (inversely proportional to the coherence time of the oscillation) and strength (the peak’s area is proportional to the variance of the QPO signal). The variance is nearly always reported in

3 terms of the root-mean-square of the signal expressed as a fraction of the count rate, the “fractional rms amplitude” r; the coherence often in terms of a quality factor Q = ν/λ. Conventionally, to call a local maximum in a power spectrum a QPO peak one requires Q> 2. Time delays between signals simultaneously detected in different energy bands are usually measured using cross-spectra (the frequency-domain equivalent of the cross-correlation function; van der Klis et al. 1987, Vaughan et al. 1994a, Nowak et al. 1998) and often expressed in terms of a phase lag (time lag multiplied by frequency). The signal-to-noise of a broad power-spectral feature is nσ = 1 2 1/2 2 Ixr (T/λ) (van der Klis 1989b, see van der Klis 1998 for more details), where I is the count rate and T the observing time (assumed 1/λ). Note x ≫ that nσ is proportional to the count rate and to the signal amplitude squared, so that it is sufficient for the amplitude to drop by 50% for the signal-to-noise to go from, e.g., a whopping 6σ to an undetectable 1.5σ – i.e., if a power-spectral feature “suddenly disappears” it may have only decreased in amplitude by a factor of two.

3 Millisecond pulsations

An accreting millisecond pulsar in a low-mass X-ray binary has sometimes been called the “Holy Grail” of X-ray astronomy. Its discovery was antici- pated for nearly 20 years, because magnetospheric disk accretion theory as well as evolutionary ideas concerning the genesis of millisecond radio pulsars strongly suggested that such rapid spin frequencies must occur in accreting low-magnetic field neutron stars (see Bhattacharya & van den Heuvel 1991). However, in numerous searches of X-ray binary time series (e.g., Leahy et al. 1983, Mereghetti & Grindlay 1987, Wood et al. 1991, Vaughan et al. 1994b) such rapid pulsars did not turn up. More than two years after RXTE’s launch, the first, and as of this writing only accreting millisecond pulsar was finally discovered on April 13, 1998 in the soft X-ray transient SAX J1808.4 3658 (Fig. 1; Wijnands & van der Klis 1998a,b). The pulse frequency is 401− Hz, so this is a 2.5 millisecond pulsar. The object is nearly certainly the same as the transient that burst out at the same position in September 1996 and gave the object its name (in ’t Zand et al. 1998). As this transient showed two type 1 X-ray bursts, SAX J1808.4 3658 is also the first genuine bursting pulsar, breaking the long-standing rule− (e.g., Lewin & Joss 1981) rule that pulsations and type 1 X-ray bursts are mutually exclusive. The orbital period of this pulsar is 2 hrs (Fig. 2; Chakrabarty & Morgan 1998a,b). With a projected orbital radius a sin i of only 63 light milliseconds

4 Figure 1: The discovery power spectrum and pulse profile (inset) of the first accreting millisecond X-ray pulsar. Note the low harmonic content evident both from the absence of harmonics in the power spectrum and the near-sinusoidal pulse profile. (after Wijnands & van der Klis 1998b)

−5 and a mass function of 3.8 10 M⊙, the companion star is either very low mass, or we are seeing the orbit nearly pole-on. The amplitude of the pulsations varied between 4 and 7% and showed little dependence on photon energy (Cui et al. 1998b). However, the pulsations as measured at higher photon energies preceded those measured in the 2-3 keV band by a gradually increasing time interval from 20 µsec (near 3.5keV) up to 200 µsec (between 10 and 25keV) (Cui et al. 1998b). These lags could be caused by Doppler shifting of emission from the pulsar hot spots, higher-energy photons being emitted earlier in the spin cycle as the spot approaches the observer (Ford 1999).

An accreting magnetized neutron star spinning this fast must have a weak magnetic field. If not, the radius of the magnetosphere rM would exceed the corotation radius, and matter corotating in the magnetosphere would not be able to overcome the centrifugal barrier. A simple estimate leads to upper 8 limits on rM of 31 km, and on the surface field strength B of 2–610 Gauss (Wijnands & van der Klis 1998b). A similarly simple estimate, involving in addition the requirement that for pulsations to occur rM must be larger than the radius R of the neutron star, would set a strong constraint on the star’s mass-radius relation (Burderi & King 1998) and hence on the EOS (see also Li et al. 1999a). However, the process of accretion onto a neutron star with such a low B is not identical to that in classical, 1012-Gauss accreting pulsars. In particular, the disk model (used in calculating rM ) is different this close to the neutron star, the disk-star boundary layer may be different, and multipole components in the magnetic field become important. Conceivably a classical magnetosphere does not even form and the 4–7%-amplitude pulsation occurs due to milder effects of the magnetic field on either the flow or the emission.

5 Figure 2: Left: the radial velocity curve and orbital elements of SAX J1808.4−3658. (Chakrabarty & Morgan 1998b) Right: the position of SAX J1808.4−3658 in the radio- pulsar period vs. magnetic-field diagram. (Psaltis & Chakrabarty 1999)

Psaltis & Chakrabarty (1999) discuss these issues and conclude that B is (1– 10)108 Gauss, which puts the source right among the other, rotation-powered, millisecond pulsars (the msec radio pulsars; Fig. 2). When the accretion shuts off sufficiently for the radio pulsar mechanism to operate, the system will likely show up as a radio pulsar. This should happen at the end of the system’s life as an X-ray binary, i.e., SAX J1808.4 3658 is indeed the long-sought millisec- ond radio-pulsar progenitor, but might− also occur in between the transient outbursts (Wijnands & van der Klis 1998b). So far, radio observations have not detected the source in X-ray quiescence (Gaensler et al. 1999). It is not clear what makes the neutron star spin detectable in SAX J1808.4 3658 and not (so far) in other low-mass X-ray binaries of similar and often much− higher flux. Perhaps a peculiar viewing geometry (e.g., a very low inclination of the binary orbit) allows us to see the pulsations only in this system, although possible X-ray and optical modulations with binary phase make an inclination of zero unlikely (Chakrabarty & Morgan 1998b, Giles et al. 1999). With a neutron star spin frequency that is certain and good estimates of rM and B, SAXJ1808.4 3658 can serve as a touchstone in studies of low- mass X-ray binaries. Although− no burst oscillations ( 4) or kHz QPOs ( 5) have been detected from the source, their absence is§ consistent with what§ would be expected from a standard LMXB in the same situation (Wijnands & van der Klis 1998c), and in more intensive observations during a next transient

6 outburst such phenomena could be detected. This would strongly test the main assumptions underlying the models for these phenomena. The X-ray spectral properties (Heindl & Smith 1998, Gilfanov et al. 1998) and the slower types of variability (Wijnands & van der Klis 1998c) of the source are very similar to those of other LMXBs at low accretion rate suggesting that either the neutron stars in those systems have similar B, or the presence of a small magnetosphere does not affect spectral and slow-variability characteristics.

4 Burst oscillations

Type 1 X-ray bursts are thermonuclear runaways in the accreted matter on a neutron-star surface (Lewin et al. 1995b for a review). When density and temperature in the accumulated nuclear fuel approach the ignition point, the matter ignites at one particular spot, from which a nuclear burning front then propagates around the star (Bildsten 1998b for a review). This leads to a burst of X-ray emission with a rise time of typically <1s, and a 101–102 s exponential decay due to cooling of the neutron-star atmosphere. The total amount of energy emitted is 1039−40 erg. In some bursts the Eddington critical luminosity is exceeded and atmospheric layers are lifted off the star’s surface, leading to an increase in photospheric radius of 101–102 km, followed by a gradual recontraction. These bursts are called “radius∼ expansion bursts”. In the initial phase, when the burning front is spreading, the energy gen- eration is inherently very anisotropic. The occasional occurrence of multiple bursts closely spaced in time indicates that not all available fuel is burned up in each burst, suggesting that in some bursts only part of the surface partici- pates. Magnetic fields and patchy burning (Bildsten 1995) could also lead to anisotropic emission during X-ray bursts. Anisotropic emission from a spin- ning neutron star leads to periodic or quasi-periodic observable phenomena, because due to the stellar rotation the viewing geometry of the brighter regions periodically varies (unless the pattern is symmetric around the rotation axis). Searches for such periodic phenomena during X-ray bursts were performed by various groups (Mason et al. 1980, Skinner et al. 1982, Sadeh et al. 1982, Sadeh & Livio 1982a,b, Murakami et al. 1987, Schoelkopf & Kelley 1991, Jongert & van der Klis 1996), but claims of detections remained unconfirmed. The first incontestable type 1 burst oscillation was discovered with RXTE in a burst that occurred on 1996 February 16 in the reliable burst source 4U1728 34. An oscillation with a slightly drifting frequency near 363Hz was evident− in a power spectrum of 32 s of data starting just before the onset of the burst (Strohmayer et al. 1996a,b,c; Fig. 3). The oscilation frequency increased from 362.5 to 363.9Hz in the course of 10s. ∼ 7 Figure 3: Left: a burst profile and its power spectrum (inset) showing a drifting burst oscillation in 4U 1728−34. (Strohmayer et al. 1996c) Right: dynamic power spectra of burst oscillations in two bursts separated by 1.6 yr in 4U 1728−34 showing near-identical asymptotic frequencies. (Strohmayer et al. 1998b)

Burst oscillations have now been detected in six (perhaps seven) different sources (Table 1). They do not occur in each burst, and some burst sources have not so far shown them at all. Sometimes the oscillations are strong for less than a second during the burst rise, then become weak or undetectable, and finally occur for 10 s in the burst cooling tail. This can happen even in radius expansion bursts,∼ where after the photosphere has recontracted the oscillations (re)appear (Smith et al. 1997, Strohmayer et al. 1997a, 1998a). Some oscillations are seen only in the burst tail and not in the rise (Smith et al. 1997). Usually, the frequency increases by 1–2 Hz during the burst tail, converg- ing to an “asymptotic frequency” which in a given source tends to be stable ((Strohmayer et al. 1998c; Fig. 3), with differences from burst to burst of 0.1% (Table 1). This is of the order of what would be expected from binary orbital∼ Doppler shifts and strengthens an interpretation in terms of the neu- tron star spin. Perhaps the orbital radial velocity curve can be detected in the asymptotic frequencies, but it remains to be seen if the asymptotic frequencies are intrinsically stable enough for this. Exceptions from the usual frequency evolution pattern do occur. Oscillations with no evidence for frequency evolu- tion were observed in KS1731 26 (Smith et al. 1997) and 4U1743 29 (Stroh- mayer et al. 1997a), and in 4U1636− 53 a decrease in frequency was− seen in a burst tail (Miller 1999b, Strohmayer− 1999; Fig. 4). In a widely (but not universally; 7.2) accepted scenario, the burst oscil- § 8 Table 1: Burst oscillations Source Minimum Asymptotic References observed frequency frequency range (Hz) (Hz) 4U 1636−53 579.3 a 581.47±0.01 581.75±0.13 1,2,3,4,5,15 4U 1702−43 329.0 329.8±0.1 330.55±0.02 6,14 4U 1728−34 362.1 363.94±0.05 364.23±0.05 2,6,7,8 KS 1731−260 523.9 523.92±0.05 9,16 MXB 1743−29b 588.9 589.80±0.07 10,17 Aql X-1 547.8 548.9 11,12,18 Rapid Burster 154.9; 306.6c 13 a A weak subharmonic exists near 290 Hz (Miller 1999a). b Source identification uncertain. c Marginal detections. References: [1] Strohmayer et al. 1998a [2] Strohmayer et al. 1998b [3] Strohmayer 1999 [4] Miller 1999a [5] Miller 1999b [6] Strohmayer & Markwardt 1999 [7] Strohmayer et al. 1996b,c [8] Strohmayer et al. 1997b [9] Smith et al. 1997 [10] Strohmayer et al. 1997a [11] Zhang et al. 1998c [12] Ford 1999 [13] Fox et al. 1999 [14] Markwardt et al. 1999a [15] Zhang et al. 1997a [16] Morgan and Smith 1996 [17] Strohmayer et al. 1996d [18] Yu et al. 1999 lations arise due to a hot spot or spots in an atmospheric layer of the neutron star rotating slightly slower than the star itself because it expanded by 5–50m in the X-ray burst but conserved its angular momentum (Strohmayer et al. 1997a and references therein, Bildsten 1998b, Strohmayer 1999, Strohmayer & Markwardt 1999, Miller 1999b). The frequency drifts are caused by spin-up of the atmosphere as it recontracts in the burst decay. The asymptotic fre- quency corresponds to a fully recontracted atmosphere and is closest to the true neutron star spin frequency. From this scenario one expects a frequency drop during the burst rise, but no good evidence has been found for this as yet (Strohmayer 1999). The case of a frequency drop in the burst tail (Fig. 4) is explained by invoking additional thermonuclear energy input late in the burst, which also affects the burst profile (Strohmayer 1999). If the oscillations are due to a stable pattern in the spinning layer, then it should be possible to describe them as a frequency-modulated, strictly coher- ent signal. By applying a simple exponential model to the frequency drifts, it is possible to establish coherences of up to Q 4000 (Strohmayer & Markwardt 1999, see also Zhang et al. 1998c, Smith et al.∼ 1997, Miller 1999a,b). However, this is still 20% less than a fully coherent signal of this frequency and dura- tion, i.e., it∼ has not yet been possible to count the exact number of cycles in the way this can be done in a pulsar. Possibly, exact coherence recovery is feasible for these signals, but current signal-to-noise limits prevent to accomplish this as the exact frequency drift ephemeris can not be found. The harmonic content of the oscillations is low. In 4U 1636 53 it is just − 9 Figure 4: Dynamic power spectra of burst oscillations (contours) overlaid on burst flux profiles (traces). Left: strong burst oscillation in 4U 1702−43 showing the usual asymptotic increase in frequency. (Strohmayer & Markwardt 1999) Right: oscillation in 4U 1636−53 exhibiting a drop in frequency in the burst tail. (Strohmayer 1999) possible, by combining data from the early stages of several bursts, to detect a frequency near 290 Hz, half the dominant one (Miller 1999a,b), suggesting that 290 Hz, not 580 Hz is the true spin frequency and that two antipodal hot ∼spots produce∼ the burst oscillation in this source (whose kHz peak separation is 250 Hz, 5). No harmonics or “subharmonics” have been seen in other sources,∼ or in§ the burst tails of any source (Strohmayer & Markwardt 1999), with the possible exception of the marginal detections in the Rapid Burster (Fox et al. 1999). The oscillation amplitudes range from 50% (rms) of the burst flux early in some bursts to between 2 and 20% (rms)∼ in the tail (references see Table 1). (Note, that sometimes sinusoidal amplitudes are reported, which are a factor √2 larger than rms amplitudes, and that amplitudes are expressed as a frac- tion of burst flux, total flux minus the persistent flux before the burst. Early in the rise, when burst flux is low compared to total flux, the measured am- plitude is multiplied by a large factor to convert it to burst flux fraction.) In KS1731 260 (Smith et al. 1997) and 4U1743 29 (Strohmayer et al. 1997a) the photon− energy dependence of the oscillations− was measured. Above 7 or 8 keV the amplitude was 9–18% while below that energy it was undetectable at <2–4%. 4U1636 53 shows a slight variation in spectral hardness as a func- tion of oscillation phase− (Strohmayer et al. 1998). In Aql X-1 photons below 5.7 keV lag those at higher energies by roughly 0.3 msec, which may be caused by Doppler shifts (Ford 1999). If the burst oscillations are due to hot spots on the surface, then their

10 amplitude constrains the “compactness” of the neutron star, defined as RG/R 2 where R is the star’s radius and RG = GM/c its gravitational radius (Stroh- mayer et al. 1997b, 1998a, Miller & Lamb 1998). The more compact the star, the lower the oscillation amplitude, as gravitational light bending increasingly blurs the beam. In particular when the oscillations are caused by two antipo- dal hot spots (cf. 4U1636 53, above, and possibly other sources, 5.5), and the amplitudes are high, the− constraints are strong. The exact bounds§ on the compactness depend on the emission characteristics of the spots, and no final conclusions have been reached yet. Modeling of the spectral and amplitude evolution of the oscillations through the burst in terms of an expanding, cooling hot spot has been succesful (Strohmayer 1997b). However, several issues with respect to this attractively simple interpretation have yet to be resolved. The presence of two burning sites as required by the description in terms of two antipodal hot spots could be related to fuel accumulation at the magnetic poles (Miller 1999a), but their simultaneous ignition seems not easy to accomplish, and obviously, in view of the frequency drifts, these sites must decouple from the magnetic field after ignition. The hot spots must survive the strong shear in layers that, from the observed phase drifts, must revolve around the star several times during the lifetime of the spots, and they must even survive through photospheric radius expansion by factors of at least several during radius-expansion bursts (which probably implies the roots of the hot spots are below the photospheric layers). The resolution of these issues ties in with questions such as why only some burst sources show the oscillations, and why only some bursts exhibit them. This is hard to explain in a magnetic-pole accumulation scheme as the viewing geometry of the poles remains the same from burst to burst. Studies of the relation between the characteristics of the bursts and the surrounding persistent emission and the presence and character of the burst oscillations could help to shed light on these various questions.

5 Kilohertz quasi-periodic oscillations

The kilohertz quasi-periodic oscillations (kHz QPOs) were discovered at NASA’s Goddard Space Flight Center in February 1996, just two months after RXTE was launched (in Sco X-1: van der Klis et al. 1996a,b,c, and 4U1728 34: Strohmayer et al. 1996a,b,c; see van der Klis 1998 for a histori- cal account).− Two simultaneous quasi-periodic oscillation peaks (“twin peaks”) in the 300–1300Hz range and roughly 300Hz apart (Fig.5) occur in the power spectra of low-mass X-ray binaries containing low-magnetic-field neutron stars of widely different X-ray luminosity Lx. The frequency of both peaks usually

11 increases with X-ray flux ( 5.4). In 4U1728 34 the separation frequency of § − the two kHz peaks is close to νburst ( 4; Strohmayer et al. 1996b,c). This com- mensurability of frequencies provides§ a powerful argument for a beat-frequency interpretation ( 5.1, 5.5, 7.1). §

Figure 5: Twin kHz peaks in Sco X-1 (left; van der Klis et al. 1997b) and 4U 1608−52 (right; M´endez et al. 1998b).

5.1 Orbital and beat frequencies Orbital motion around a neutron star occurs at a frequency of

1/2 −3/2 GM rorb 1/2 νorb = 2 3 1200 Hz m1.4 , 4π rorb  ≈ 15km and the corresponding orbital radius is

1/3 −2/3 GM νorb 1/3 rorb = 2 2 15km m1.4 , 4π νorb  ≈ 1200 Hz where m1.4 the star’s mass in units of 1.4M⊙ (Fig.6). In general relativ- ity, no stable orbital motion is possible within the innermost stable circu- 2 lar orbit (ISCO), RISCO = 6GM/c 12.5m1.4 km. The frequency of or- bital motion at the ISCO, the highest≈ possible stable orbital frequency, is νISCO (1580/m1.4) Hz. These≈ expressions are valid for a Schwarzschild geometry, i.e., outside a non-rotating spherically symmetric neutron star (or black hole). Corrections to first order in j = cJ/GM 2, where J is the neutron-star angular momentum have been given by, e.g., Miller et al. (1998a) and can be several 10%. For more precise calculations see Morsink & Stella (1999). In spin-orbit beat-frequency models some mechanism produces an interac- tion of νorb at some preferred radius in the accretion disk with the neutron star

12 1200 Hz 500 Hz

Figure 6: A 10-km radius, 1.4M⊙ neutron star with the corresponding innermost circu- lar stable orbit (ISCO; dashed circles) and orbits (drawn circles) corresponding to orbital frequencies of 1200 and 500 Hz, drawn to scale. spin frequency ν , so that a beat signal is seen at the frequency ν = ν ν . s beat orb− s As νbeat is the frequency at which a given particle orbiting in the disk over- takes a given point on the spinning star, it is the natural disk/star interaction frequency. In such a rotational interaction (with spin and orbital motion in the same sense) no signal is produced at the sum frequency νorb + νs (it is a “single-sideband” interaction).

5.2 Early interpretations

It was immediately realized that the observed high frequencies of kHz QPOs could arise in orbital motion of accreting matter very closely around the neu- tron star, or in a beat between such orbital motion and the neutron-star spin (van der Klis et al. 1996a, Strohmayer et al. 1996a; some of the proposals to look for such rapid QPOs with RXTE had in fact anticipated this). A magne- tospheric spin-orbit beat-frequency model (Alpar & Shaham 1985, Lamb et al. 1985) was already in use in LMXBs for slower QPO phenomena ( 6.3) so when three commensurable frequencies were found in 4U 1728 34, a beat-frequency§ interpretation was immediately proposed (Strohmayer et− al. 1996c). Let us call ν2 the frequency of the higher-frequency (the “upper”) peak and ν1 that of the lower-frequency kHz peak (the “lower peak”). Then the beat-frequency inter- pretation asserts that ν2 is νorb at some preferred radius in the disk, and ν1 is the beat frequency between ν and ν , so ν = ν = ν ν ν ν , 2 s 1 beat orb − s ≈ 2 − burst where the approximate equality follows from νburst νs ( 4). That only a single sideband is observed is a strong argument for≈ a rotati§ onal interaction ( 5.1). Later this was worked out in detail in the form of the “sonic point beat-frequency§ model” by Miller et al. (1996, 1998a; 7.1), where the pre- § 13 ferred radius is the sonic radius (essentially, the inner edge of the Keplerian disk). The beat-frequency interpretation implies that the observed kHz QPO peak separation ∆ν = ν ν should be equal to the neutron star spin frequency 2− 1 νs, and should therefore be constant and nearly equal to νburst. As we shall see in 5.5, it turned out that ∆ν is not actually exactly constant nor precisely § equal to νburst (in 7 we examine how a beat-frequency interpretation could deal with this), and§ this triggered the development of other models for the kHz QPOs. Stella & Vietri (1998) noted that the frequency νLF of low-frequency QPOs ( 6.3) in the 15–60Hz range that had been known in the Z sources since the 1980’s§ (cf. van der Klis 1995), and that were being discovered with RXTE 2 in the atoll sources as well is approximately proportional to ν2 . This triggered a series of papers (Stella & Vietri 1998, 1999, Stella et al. 1999b) together describing what is now called the “relativistic precession model”, where ν2 is the orbital frequency at some radius in the disk and ν1 and νLF are frequencies of general-relativistic precession modes of a free particle orbit at that radius ( 7.2). For a further discussion of kHz QPO models see 7. With the exception of§ the photon bubble model (Klein et al. 1996b; 7.3), all§ models are based on the interpretation that one of the kHz QPO frequencies§ is an orbital frequency in the disk.

5.3 Dependence on source state and type Twenty sources have nowa shown kHz QPOs. Sometimes only one peak is detectable, but 18 of these sources have shown two simultaneous kHz peaks; the exceptions with only a single peak are the little-studied XTEJ1723 376 and EXO 0748 676. Tables 2 and 3 summarize the results. There is a remarkab− le similarity− in QPO frequencies and peak separations across a great variety of sources. However, at a more detailed level there turn out to be differences between the different source types. The two main types are the Z sources and the atoll sources (Hasinger & van der Klis 1989, see van der Klis 1989a, 1995a,b for reviews). Z sources are named after the roughly Z-shaped tracks they trace out in X-ray color-color and hardness-intensity diagrams on a time scale of hours to days (Fig. 7). They are the most luminous LMXBs, with X-ray luminosity Lx near the Eddington luminosity LEdd. Atoll sources produce tracks roughly like a wide U or C (e.g., Fig. 7) which are somewhat reminiscent of a geographical map of an atoll because in the left limb of the U the motion through the diagram becomes slow, so that the track is usually broken up by observational a1999 October 15

14 Table 2: Observed frequencies of kilohertz QPOs in Z sources

Source ν1 ν2 ∆ν νburst References (Hz) (Hz) (Hz) (Hz) Sco X-1 565 870 307±5 Van der Klis et al. 1996a,b,c,1997b 845 1080 237±5 1130 − GX 5 1 215 505298±11 Van der Klis et al. 1996e 660 890 Wijnands et al. 1998c 700 GX 17+2 645 Van der Klis et al. 1997a 480 785294±8 Wijnands et al. 1997b 780 1080 Cyg X-2 730 Wijnands et al. 1998a 530 855 346±29 660 1005 GX 340+0 200 535339±8 Jonker et al. 1998, 1999c 565 840 625 GX 349+2 710 980 266±13 Zhang et al. 1998a a 1020 Kuulkers & van der Klis 1998

Values for ν1 and ν2 were rounded to the nearest 5, for νburst to the nearest 1Hz. Entries in one column not separated by a horizontal line indicate ranges over which the frequency was observed or inferred to vary; ranges from different observations were combined assuming the ν1, ν2 relation in each source is reproducible (no evidence to the contrary exists). Entries in one uninterrupted row refer to simultaneous data (except for νburst values). Values of ∆ν straddling two rows, or adjacent to a vertical line refer to measurements made over the range of frequencies indicated. a Note: Marginal detection. windowing into “islands”. The bottom and right-hand parts of the U are traced out in the form of a curved “banana” branch on a time scale of hours to a day. In a given source, the islands correspond to lower flux levels than the banana branch. Most atoll sources are in the 0.01–0.2 LEdd range; a group of four bright ones (GX3+1, GX9+1, GX13+1 and GX9+9) that is nearly always in the banana branch is more luminous than this, perhaps 0.2–0.5LEdd (the distances are uncertain). Most timing and spectral characteristics of these sources depend in a simple way on position along the Z or atoll track. So, the phenomenology is essentially one-dimensional. A single quantity, usually referred to as “inferred accretion rate”, varying on time scales of hours to days on the Z track and the banana branch, and more slowly in the island state must govern most of the phenomenology (but see 5.4). § In all Z sources and in 4U1728 34 the kHz QPOs are seen down to the lowest inferred M˙ levels these sources− reach. The QPOs always become unde- tectable at the highest M˙ levels. In the atoll sources, where the count rates are higher at higher inferred M˙ , this can not be a sensitivity effect. In most atoll sources, the QPOs are seen in the part of the banana branch closest to the islands, i.e., near the lower left corner of the U (Fig. 7); that they are often not detected in the island state may be related to low sensitivity at the low count rates there, but in one island in 4U 0614+09 the undetected lower kHz

15 Table 3: Observed frequencies of kilohertz QPOs in atoll sources ν ν ν ν Source 1 2 ∆ burst References d (Hz) (Hz) (Hz) (Hz) 4U 0614+09 450 Ford et al. 1996, 1997a,b; Van der Klis et 418 765 al. 1996d; M´endez et al. 1997; Vaughan 312±2 et al. 1997,1998; Kaaret et al. 1998; van 825 1160 Straaten et al. 1999 1215 1330 EXO 0748−676 695 J. Homan 1999, in prep. 4U 1608−52 415 Van Paradijs et al. 1996; Berger et al. 440 765 325±7 1996; Yu et al. 1997; Kaaret et al. 1998; Vaughan et al. 1997,1998; M´endez et al. 475 800 326±3 1998a,b, 1999; M´endez 1999; Markwardt b b 865 1090 225±12 et al. 1999b 895 4U 1636−53 830 Zhang et al. 1996a,b,1997a; Van der Klis 900 1150 et al. 1996d; Vaughan et al. 1997,1998; b Zhang 1997; Wijnands et al. 1997a; 950 1190 251±4 291,582 M´endez et al. 1998c; M´endez 1999; 1230 Markwardt et al. 1999b; Kaaret et al. 1999a 1070 4U 1702−43 625 Markwardt et al. 1999a,b b b 655 1000 344±7 b b 700 1040 337±7 330 b b 770 1085 315±11 902 a 4U 1705−44 775 1075 298±11 Ford et al. 1998a 870 XTE J1723−376 815 Marshall & Markwardt 1999 4U 1728−34 325 Strohmayer et al. 1996a,b,c; Ford & van 510 845 der Klis 1998; M´endez & van der Klis c 1999; M´endez 1999; Markwardt et al. 875 1160 349±2 364 c 1999b; di Salvo et al. 1999 920 279±12 KS 1731−260 900 1160 260±10 524 Wijnands & van der Klis 1997 1205

4U 1735−44 630 980 341±7 Wijnands et al. 1996a, 1998b; Ford et al. 730 1025 296±12 1998b a 900 1150 249±15 1160 4U 1820−30 655 Smale et al. 1996, 1997; Zhang et al. 500 860 358±42 1998b; Kaaret et al. 1999b; Bloser et al. 1999 795 1075 278±11 1100 Aql X-1 670 a a Zhang et al. 1998c; Cui et al. 1998a; Yu 1040 241±9 930 549 et al. 1999; Reig et al. 1999; M. M´endez et al. 1999 in prep. 4U 1915-05 820 Barret et al. 1997,1998; Boirin et al. 515 1999 560 925 655 1005 348±11 a 705 1055 880 a 1265 XTE J2123−058 845 1100 255±14 Homan et al. 1998b,1999a; Tomsick et al. 855 1130 276±9 1999 a a 870 1140 270±5 a b Caption: see Table 2. Notes: Marginal detection. Shift and add detection method, cf. M´endez et al. (1998a). c d See Fig. 10. For burst oscillation references see Table 1. peak is really much weaker than at higher inferred M˙ (M´endez et al. 1997). No kHz QPOs have been seen in the four bright atoll sources (Wijnands et al. 1998d, Strohmayer 1998, Homan et al. 1998a), perhaps because they do not usually reach this low part of the banana branch, and in several faint LMXBs, probably also atoll sources in the island state (SAX J1808.4 3658; Wijnands & van der Klis 1998c and 3, XTEJ1806 246; Wijnands & van− der Klis 1999b, SLX 1735 269; Wijnands§ & van der Klis− 1999c, 4U1746 37; Jonker et al. 1999a, 4U1323− 62; Jonker et al. 1999b, 1E1724 3045, SLX1735− 269 and − − − 16 Horizontal Branch

Sz =1

Normal Branch

S a = 1

S = 2 a Sz =2 Flaring Branch

Figure 7: Left: X-ray color-color diagram of the atoll source 4U 1608−52. (M´endez et al. 1999) Right: X-ray hardness vs. intensity diagram of the Z source GX 340+0. (after Jonker et al. 1999c) Values of curve-length parameters Sz and Sa and conventional branch names are indicated. Mass accretion rate is inferred to increase in the sense of increasing Sa and Sz. X-ray color is the log of a count rate ratio (3.5–6.4/2.0–3.5 and 9.7–16.0/6.4–9.7 keV for soft and hard color respectively); intensity is in the 2–16 keV band. kHz QPO detections are indicated with filled symbols.

GS 1826 238; Barret et al. 1999). − The QPO frequencies increase when the sources move along the tracks in the sense of increasing inferred M˙ (this has been seen in more than a dozen sources, and no counterexamples are known). On time scales of hours to a day increasing inferred M˙ usually corresponds to increasing X-ray flux, so on these time scales kHz QPO frequency usually increases with flux. When flux systematically decreases with inferred M˙ , as is the case in some parts of these tracks, the frequency is expected to maintain its positive correlation with inferred M˙ and hence become anticorrelated to flux, and indeed this has been observed in the Z source GX 17+2 (Wijnands et al. 1997b). So, the kHz QPOs fit well within the pre-existing Z/atoll description of LMXB phenomenology in terms of source types and states, including the fact that position on the tracks in X-ray color-color or hardness-intensity diagrams (“inferred M˙ ”, see 5.4) and not X-ray flux drives the phenomenology. §

5.4 Dependence of QPO frequency on luminosity and spectrum

Kilohertz QPOs occur at similar frequency in sources that differ in X-ray lu- minosity Lx by more than 2 orders of magnitud, and the kHz QPO frequency

17 ν seems to be determined more by the difference between average and instan- taneous Lx of a source than by Lx itself (van der Klis 1997, 1998). In a plot 2 of ν vs. Lx (defined as 4πd fx with fx the X-ray flux and d the distance; Ford et al. 1999; Fig.8) a series of roughly parallel lines is seen, to first order one line per source (but see below). In each source there is a definite relation between Lx and ν, but the same relation does not apply in another source with a different average Lx. Instead, that source covers the same ν-range around its particular average Lx. This is unexplained, and must mean that in addition to instantaneous Lx, another parameter, related to average Lx, affects the QPO frequency (van der Klis 1997, 1998). Perhaps this parameter is the neutron star magnetic field strength, which previously, on other grounds, was hypothe- sized to correlate to average Lx (Hasinger & van der Klis 1989, Psaltis & Lamb 1998, Konar & Bhattacharya 1999, see also Lai 1998), but other possibilities exist (van der Klis 1998, Ford et al. 1999).

Figure 8: The parallel lines phenomenon across sources (left; Ford et al. 1999; upper and lower kHz peaks are indicated with different symbols) and in the source 4U 1608−52 (right; M´endez et al. 1999, frequency plotted is ν1).

A similar pattern of parallel lines, but on a much smaller scale, occurs in some individual sources. When observed at different epochs, a source produces different frequency vs. flux tracks that are approximately parallel (GX 5 1: Wijnands et al. 1998c; GX340+0: Jonker et al. 1998; 4U0614+09: Ford− et al. 1997a,b, M´endez et al. 1997, van Straaten et al. 1999; 4U 1608 52: Yu et al. 1997, M´endez et al. 1998a, 1999; 4U1636 53: M´endez 1999; 4U1728− 34: M´endez & van der Klis 1999; 4U1820 30:− Kaaret et al. 1999b; AqlX-1:− − 18 Zhang et al. 1998c, Reig et al. 1999; Figs.8 and 9). This is most likely another aspect of the well-known fact (e.g., van der Klis et al. 1990, Hasinger et al. 1990, Kuulkers et al. 1994, 1996, van der Klis 1994, 1995a) that while the properties of timing phenomena such as QPOs are well correlated with one another and with X-ray spectral shape as diagnosed by X-ray colors (and hence with position in tracks in color-color diagrams) the flux correlates well to these diagnostics only on short (hours to days) time scales and much less well on longer time scales. This is why color-color diagrams, independent of flux, are popular for parametrizing spectral variability in these sources. Similarly to other timing parameters, kHz QPO frequency correlates much better with position on the track in the color-color diagram (Fig. 9) than with flux. Correlations of frequency with parameters describing spectral shape such as blackbody flux (Ford et al. 1997b) or power-law slope (Kaaret et al. 1998) in a two-component spectral model are also much better than with flux.

Figure 9: In 4U 1728−34 a QPO frequency (ν1) vs. count rate plot (left) shows no clear correlation, but instead a series of parallel lines. When the same data are plotted vs. position in the X-ray color-color diagram a single relation is observed. (after M´endez & van der Klis 1999)

Usually, all this has been interpreted by saying that apparently inferred accretion rate ( 5.3) governs both the timing and the spectral properties, but not flux (e.g., van§ der Klis 1995a). Of course, energy conservation suggests total M˙ and flux should be well correlated. Perhaps, inferred M˙ is not the total M˙ but only one component of it, i.e., that through the disk, while there is also a radial inflow (e.g. Fortner et al. 1989, Kuulkers & van der Klis

19 1995, Wijnands et al. 1996b, Kaaret et al. 1998), maybe there are large and variable anisotropies or bolometric corrections in the emission, so that the flux we measure is not representative for the true luminosity (e.g., van der Klis 1995a), or possibly mass outflows destroy the expected correlation by providing sinks of both mass and (kinetic) energy (e.g., Ford et al. 1999). The true explanation is unknown. We do not even know if the two different parallel-lines phenomena (across sources and within individual sources; Fig. 8) have the same origin. It is possible that the quantity that everything depends on is not M˙ , but some other parameter such as inner disk radius rin.

5.5 Peak separation and burst oscillation frequencies

In interpretations ( 5.1, 7.1) where the kHz peak separation ν ν = ∆ν § 2 − 1 is the neutron star spin frequency νs one expects ∆ν to be approximately constant. This is not the case (Fig. 10). In Sco X-1 (van der Klis 1996c, 1997b), 4U1608 52 (M´endez et al. 1998a,b), 4U1735 44 (Ford et al. 1998b), 4U1728 34 (M´endez− & van der Klis 1999) and marginally− also 4U1702 43 (Markwardt− et al. 1999a) the separation ∆ν decreases considerably when− the kHz QPO frequencies increase. Other sources may show similar ∆ν variations (see also Psaltis et al. 1998). In a given source the ν1, ν2 relation seems to be reproducible.

Figure 10: Left: the variations in kHz QPO peak separation as a function of the lower kHz frequency. Right: the data on 4U 1728−34. The burst oscillation frequency is indicated by a horizontal drawn line (M´endez & van der Klis 1999).

20 That νburst, by interpretation close to νs ( 4), is close to ∆ν (or 2∆ν) is the most direct evidence for a beat frequency interpretatio§ n of the kHz QPOs ( 5.1). The evidence for this is summarized in Table 4, where the highest asymptotic§ burst frequency (likely to be closest to the spin frequency; 4) is compared to the largest well-measured ∆ν in the five sources where both§ have been measured. The frequency ratio may cluster at 1 and 2, but a few more examples are clearly needed; in two sources discrepancies of 15% occur ∼ between νburst and 2∆ν. Of these five sources, 4U1728 34 also has a measured ∆ν variation (Fig. 10). When the QPO frequency drops,− ∆ν increases, to saturate 4% below the burst oscillation frequency (M´endez & van der Klis 1999). It certainly seems here as if the kHz QPO separation “knows” the value of the burst oscillation frequency.

Table 4: Commensurability of kHz QPO and burst oscillation frequencies Source Highest Highest Ratio Discrepancy References νburst ∆ν (νburst/∆ν) (Hz) (Hz) (%) 4U 1702−43 330.55±0.02 344±7 0.96±0.02 −4±2 1 4U 1728−34 364.23±0.05 349.3±1.7 1.043±0.005 +4.3±0.5 2 KS 1731−260 523.92±0.05 260±10 2.015±0.077 +0.7±3.8 3 Aql X-1 548.9 241±9a 2.28±0.09a +14±5a 4 4U 1636−53 581.75±0.13 254±5 2.29±0.04 +15±2 5 a Based on a marginal detection (see Table 3). References: [1] Markwardt et al. 1999a [2] M´endez & van der Klis 1999 [3] Wijnands & van der Klis 1997 [4] M. M´endez et al. 1999 in prep. [5] M´endez et al. 1998c

5.6 Strong gravity and dense matter Potentially kHz QPOs can be used to constrain neutron-star masses and radii, and to test general relativity. Detection of the predcited innermost stable circu- lar orbit (ISCO, 5.1) would constitute the first direct detection of a strong-field general-relativistic§ effect and prove that the neutron star is smaller than the ISCO. This possibility has fascinated since the beginning and was discussed well before the kHz QPOs were found (Klu´zniak and Wagoner 1985, Paczyn- ski 1987, Klu´zniak et al. 1990, Klu´zniak & Wilson 1991, Biehle & Blandford 1993). If a kHz QPO is due to orbital motion around the neutron star, its fre- quency can not be larger than the frequency at the ISCO. Kaaret et al. (1997) proposed that kHz QPO frequency variations that then seemed uncorrelated to X-ray flux in 4U 1608 52 and 4U1636 53 were due to orbital motion near − − the ISCO and from this derived neutron star masses of 2M⊙. However, from more detailed studies ( 5.4) it is clear now that on short∼ time scales frequency § 21 does in fact correlate to flux in these sources. The maximum kHz QPO frequencies observed in each source are con- strained to a narrow range. The 12 atoll sources with twin peaks have maxi- mum ν2 values in the range 1074–1329Hz; among the Z sources there are two cases of much lower maximum ν2 values (<900Hz in GX5 1 and GX340+0), while the other four fit in. Zhang et al. (1997b) proposed− that this narrow distribution is caused by the limit set by the ISCO frequency, which led them to neutron star masses near 2M⊙ as well. It is in principle possible that the maximum is set by some other limit on orbital radius (e.g., the neutron star surface), or that it is caused by the ISCO, but the frequency we observe is not orbital, in which cases no mass estimate can be made.

Figure 11: Left: evidence for a leveling off of the kHz QPO frequency with count rate in 4U 1820−30. (Zhang et al. 1998b) Right: a larger data set (see also Kaaret et al. 1999b) of 4U 1820−30 at higher resolution showing only the lower peak’s frequency ν1. (M. M´endez et al. in prep.).

Miller et al. (1996, 1998a) suggested that when the inner edge of the accretion disk reaches the ISCO, the QPO frequency might level off and re- main constant while M˙ continues rising. Later, an apparent leveling off at ν2=1060 20Hz with X-ray count rate was found in 4U 1820 30 (Zhang et al. 1998b; Fig.± 11, left). If this is the orbital frequency at the−ISCO, then the neutron star has a mass mass of 2.2M⊙ and is smaller than its ISCO, and many equations of state are rejected.∼ The leveling off is also observed as a function of X-ray flux and color (Kaaret et al. 1999b) and position along the

22 atoll track (Bloser et al. 1999). However, the frequency vs. flux relations are known in other sources ( 5.4) to be variable, and in 4U1820 30 the leveling off seems not to be reproduced§ in the same way in all data sets− (M. M´endez et al. in prep.; Fig. 11, right). It may also be more gradual in nature than originally suggested. No evidence for a similar saturation in frequency was seen in other sources, and most reach higher frequencies. Possibly, the unique aspects of the 4U1820 30 system are related to this (it is an 11-min binary in a globular cluster with− probably a pure He companion star; Stella et al. 1987a). If a kHz QPO peak at frequency ν corresponds to stable Keplerian motion around a neutron star, one can immediately set limits on neutron star mass M and radius R (Miller et al. 1998a). For a Schwarzschild geometry: (1) the radius R of the star must be smaller than the radius rK of the Keplerian orbit: R < r = (GM/4π2ν2)1/3, and (2) the radius of the ISCO ( 5.1) K § must also be smaller than rK , as no stable orbit is possible within this radius: 2 2 2 1/3 3 3/2 rISCO =6GM/c < (GM/4π ν ) or M

If ∆ν is near νs then the 18 neutron stars where this quantity has been measured all spin at frequencies between 240 and 360 Hz, a surprisingly nar- row range. If the stars spin at the magnetospheric∼ ∼ equilibrium spin rates (e.g., Frank et al. 1992) corresponding to their current luminosities Lx, this would imply an unlikely, tight correlation between Lx and neutron-star magnetic- field strength (White & Zhang 1997; note that a similar possibility came up in the discussion about the uniformity of the QPO frequencies themselves, 5.4). § White & Zhang (1997) propose that when rM is small, as is the case here, it

23 Figure 12: Constraints on the mass and radius of neutron stars from the detection of or- bital motion with the frequencies indicated. Graphs are for negligible neutron star angular momentum (left) and for the values of j = cJ/GM 2 (§5) indicated (right). Mass-radius relations for some representative EOSs are shown. (Miller et al. 1998a) depends only weakly on accretion rate (as it does in some inner disk models; cf. Ghosh & Lamb 1992, Psaltis & Chakrabarty 1999). Another possibility is that the spin frequency of accreting neutron stars is limited by gravitational radiation losses (Bildsten 1998a, Andersson et al. 1999, see also Levin 1999). If so, then gravitational radiation is transporting angular momentum out as fast as accretion is transporting it in, making these sources the brightest gravi- tational radiation sources in the sky. They would produce a periodic signature at the neutron star spin frequency, which would facilitate their detection.

5.7 Other kilohertz QPO properties

The amplitudes of kHz QPOs increase strongly with photon energy (Fig. 13). In similar X-ray photometric bands the QPOs tend to be weaker in the more luminous sources, with 2–60 keV amplitudes ranging from as high as 15% (rms) in 4U0614+09, to typically a few % (rms) at their strongest in the Z sources. At high energy amplitudes are much higher (e.g., 40% rms above 16 keV in 4U0614+09; M´endez et al. 1997). Fractional rms usually decreases with in- ferred M˙ (e.g., van der Klis et al. 1996b, 1997b, Berger et al. 1996) but more complex behavior is sometimes seen (e.g., di Salvo et al. 1999). The measured widths of QPO peaks are affected by variations in centroid frequency during the measurement, but typical values in the Z sources are 50–200Hz. In the

24 atoll sources, the upper peak usually has a width similar to this, although occasionally peaks as narrow as 10 or 20Hz have been measured (eg. Wij- nands & van der Klis 1997, Wijnands et al. 1998a,b) but the lower peak is clearly much narrower. It is rarely as wide as 100 Hz (M´endez et al. 1997) and sometimes as narrow as 5Hz (e.g., Berger et al. 1996, Wijnands et al. 1998b). Various different peak width vs. inferred M˙ relations have been seen, but there seems to be a tendency for the upper peak to become narrower as its frequency increases.

Figure 13: Energy dependence (left) and time lags (right) of the lower kHz peak in 4U 1608−52. (Berger et al. 1996, Vaughan et al. 1997, 1998)

A strong model constraint is provided by the time lags between kHz QPO signals in different energy bands ( 2). Time-lag measurements require very high signal-to-noise ratios (Vaughan§ et al. 1997), and have mostly been made in the very significant lower peaks observed in some atoll sources. Finite lags of 10–60µsec occur in these peaks (Vaugan et al. 1997, 1998, Kaaret et al. 1999a, Markwardt et al. 1999b; see Lee & Miller 1998 for a calculation of Comptonization lags relevant to kHz QPOs). Contrary to the initial report, the low-energy photons lag the high-energy ones (these are “soft lags”) by increasing amounts as the photon energy increases (Fig. 13). The lags are of opposite sign to those expected from the inverse Compton scattering thought to produce the hard spectral tails of these sources (e.g., Barret & Vedrenne 1994), and correspond to light travel distances of only 3–20 km. From this it seems more likely that the lags originate in the QPO production mechanism than in propagation delays. Markwardt et al. (1999b) reported a possible hard lag in an atoll-source upper peak.

25 6 Correlations with low-frequency timing phenomena and with other sources 6.1 Black hole candidates

The only oscillations with frequencies exceeding 102.5 Hz ( 1) known in black hole candidates (BHCs) are, marginally, the 100–300 Hz§ oscillations in GRO J1655 40, and XTEJ1550 564, and those reported very recently in CygX-1 and− 4U1630 47. An oscillation− near 67Hz observed in GRS1915+105 is usually discussed− together with these QPOs, although it is not clear that it is related. Usually, the phenomenology accompanying these high-frequency QPOs (spectral variations, lower-frequency variations) is complex. The 67Hz QPO in GRS1915+105 (Morgan et al. 1997, Remillard & Mor- gan 1998) varies by only a few percent in frequency when the X-ray flux varies by a factor of several. It is relatively coherent, with Q usually around 20 (but sometimes dropping to 6; Remillard et al. 1999a), has an rms amplitude of about 1%, and the signal at high photon energy lags that at lower energy by up to 2.3 radians (Cui 1999). The 300Hz QPO in GROJ1655 40 (Remillard et al. 1999b) was seen only when the X-ray spectrum was dominated− by a hard power law component. This feature was relatively broad (Q 4) and weak (0.8% rms), and did not vary in frequency by more than 30Hz.∼ The 185–285Hz QPO in XTEJ1550 564 (Remillard et al. 1999c) shows∼ considerable variations in frequency (Homan− et al. 1999b, Remillard et al. 1999c). It is seen in similar X- ray spectral conditions as the 300Hz QPO in GROJ1655 40, has similarly low amplitudes and 3

26 work clearing up the exact phenomenology and more observations of black- hole transients leading to more examples of high frequency QPOs are clearly needed.

6.2 Cen X-3 The detection of QPO features near 330 and 760Hz in the 4.8s accreting pulsar CenX-3 was recently reported by Jernigan et al. (1999). This is the first report of millisecond oscillations from a high-magnetic-field ( 1012 Gauss) neutron star. The QPO features are quite weak. Jernigan et∼ al. (1999) carefully discuss the instrumental effects, which are a concern at these low power levels and interpret their results in terms of the photon bubble model ( 7.3). § 6.3 Low-frequency phenomena Low-frequency (<100 Hz) QPOs have been studied in accreting neutron stars and black hole candidates since the 1980’s, mostly with the EXOSAT and Ginga satellites (see van der Klis 1995a). Two different ones were known in the Z sources, the 6-20Hz so-called normal and flaring-branch oscillation (NBO; Middleditch & Priedhorsky 1986) and the 15–60 Hz so-called horizontal branch oscillation (HBO; van der Klis et al. 1985). The frequency of the HBO turned out to correlate well to those of the kHz QPOs (Fig. 15; van der Klis et al. 1997b, Wijnands et al. 1997b,1998a,c, Jonker et al. 1998, 1999c), and the same is true for the NBO in Sco X-1 (van der Klis et al. 1996b). Broad power- spectral bumps and, rarely, low-frequency QPOs were know in atoll sources as well (Lewin et al. 1987, Stella et al. 1987b, Hasinger & van der Klis 1989, Dotani et al. 1989, Makishima et al. 1989, Yoshida et al. 1993). With RXTE, QPOs similar to HBO are often seen (Strohmayer et al. 1996b, Wijnands & van der Klis 1997, Wijnands et al. 1998b, Homan et al. 1998a) and their frequencies also correlate well to kHz QPO frequency (Fig. 15; Stella & Vietri 1998, Ford & van der Klis 1998, Markwardt et al. 1999a, van Straaten et al. 1999, di Salvo et al. 1999, Boirin et al. 1999). It is not sure yet wether these atoll QPOs are physically the same as HBO in Z sources but this seems likely. It is possible that these correlations arise just because all QPO phenomena depend on a common parameter (e.g. inferred M˙ , 5.4), but Stella & Vietri (1998) proposed that their origin is a physical dependence§ of the frequencies on one another ( 7.2). In their relativistic precession model the HBO and the similar-frequency§ QPOs in the atoll sources are the same phenomenon, and 2 their frequency νLF is predicted to be proportional to ν2 . This is indeed approximately true in all Z sources except Sco X-1 (Psaltis et al. 1999b), as well as in the atoll sources (references above). Psaltis et al. (1999b) argue that

27 QPO QPO

Break Break

SAX J1808.4-3658 4U 1728-34

QPO

Break QPO

Break

Cygnus X-1 GX 340+0

Figure 14: Broad-band power spectra of, respectively, the millisecond pulsar, an atoll source, a black-hole candidate and a Z source. (Wijnands & van der Klis 1999a) a combination of the sonic-point and magnetospheric beat-frequency models can explain these correlations as well ( 7.1). § Additional intriguing correlations exist between kHz QPOs and low- frequency phenomena which may link neutron stars and BHCs. It is useful to first examine a correlation between two low-frequency phenomena. At low M˙ , BHCs and atoll sources (van der Klis 1994a and references therein), the millisecond pulsar SAX J1808.4-3658 ( 3; Wijnands & van der Klis 1998c), and perhaps even Z sources have very similar§ power spectra (Fig. 14; Wijnands & van der Klis 1999a), with a broad noise component that shows a break at low frequency and often a QPO-like feature above the break. Break and QPO frequency both vary in excellent correlation (Fig. 15, left), and similarly in neutron stars and BHCs. This suggests that (with the possible exception of the Z sources, which are slightly off the main relation) these two phenomena are the same in neutron stars and black holes. This would exclude spin-orbit beat-frequency models and any other models requiring a material surface, an event horizon, a magnetic field, or their absence, and would essentially imply the phenomena are generated in the accretion disk around any low-magnetic field compact object.

28 Figure 15: The frequencies of various QPO and broad noise components seen in accreting neutron stars and black holes plotted vs. each other suggest that components similar to Z-source HBO (νLF ) and the lower kHz peak (ν1) occur in all these sources and span a very wide range in frequency. Left: νLF vs. noise break frequency (after Wijnands & van der Klis 1999a); right: νLF vs. ν1 (after Psaltis et al. 1999a). Filled circles represent black hole candidates, open circles Z sources, crosses atoll sources (the smaller crosses in the right hand frame are data of 4U 1728−34 where ν1 was obtained from ν1 = ν2 − 363 Hz), triangles the millisecond pulsar SAX J1808.4−3658, pluses faint burst sources and squares (from Shirey et al. 1996) Cir X-1.

The good correlations between kHz QPOs and low-frequency phenomena in Z and atoll sources suggest that kHz QPOs might also fit in with schemes linking neutron stars and BHCs ( 6.1). However, no twin kHz QPOs have been reported from BHCs. Psaltis et§ al. (1999a) pointed out that many Z and atoll sources, the peculiar source Cir X-1 and a few low luminosity neutron stars and BHCs sometimes show two QPO or broad noise phenomena whose centroid frequencies, when plotted vs. each other, seem to line up (Fig. 15, right). This suggests that perhaps “kHz QPOs” do occur in BHCs, but as features at frequencies below 50 Hz. These features have very low Q, and although the data are suggestive they are not conclusive. The implication would be that the lower kHz QPO peak (whose frequency is the one that lines up with those seen in the BHCs) is not unique to neutron stars, but a feature of disk accretion not related to neutron star spin. The coincidence of kHz QPO frequencies with burst oscillation frequencies ( 4, 5.1) would then require some other explanation. Orbital motion in the disk§ would§ remain an attractive interpretation for some of the observed frequencies. Stella et al. (1999b) showed that for particular choices of neutron star and black hole

29 angular momenta their relativistic precession model can fit these data. The phenomenology is quite complex; in particular, no way has been found yet to combine the Wijnands & van der Klis (1999a) work with the Psaltis et al. (1999a) results in a way that works across all source types. New low-frequency phenomena are still being discovered as well (di Salvo et al. 1999, Jonker et al. 1999c).

7 Kilohertz QPO models

The possibility to derive conclusions of a fundamental nature has led to a relatively large number of models for kHz QPOs. Most, but not all of these involve orbital motion around the neutron star. It is beyond the scope of the present work to provide an in-depth discussion of each model. Instead, I point out some of the main issues, and provide pointers to the literature. In early works, the magnetospheric beat-frequency model was implied when beat-frequency models were mentioned (e.g., van der Klis et al. 1996b, Strohmayer et al. 1996c), but this model has not recently been applied much to kHz QPOs; see however Cui et al. (1998a). Most prominent recently have been the sonic point beat-frequency model of Miller et al. (1996, 1998a) and the relativistic precession model of Stella and Vietri (1998, 1999), but also the the photon bubble model (Klein et al. 1996b) and the disk transition layer models (Titarchuk et al. 1998, 1999) have been strongly argued for. Addi- tional disk models have been proposed as well ( 7.4). Neutron star oscillations have been considered (Strohmayer et al. 1996c,§ Bildsten et al. 1998, Bildsten & Cumming 1998), but probably can not produce the required combination of high frequencies and rapid changes in frequency. Most models have evolved in response to new observational results. The sonic point model was modified to accommodate the observed deviations from a pure beat-frequency model ( 5.5; Lamb & Miller 1999, Miller 1999c), the rel- ativistic precession model initially§ explained the lower kHz peak as a spin/orbit beat frequency (Stella & Vietri 1998) and only later by apsidal motion (Stella & Vietri 1999), the photon bubble model is based on numerical simulations which in time have become more similar to what is observed (R Klein, priv. comm.), and also the disk transition layer models have experienced consider- able evolution (e.g., Osherovich & Titarchuk 1999). The relativistic precession model makes the strongest predictions with re- spect to observable quantities and hence allows the most direct tests, but the near-commensurability of kHz QPO and burst oscillation frequencies ( 5.5) is unexplained in that model. The sonic-point model provides specific mech-§ anisms to modulate the X-rays and to make the frequency vary with mass-

30 accretion rate. Other models usually discuss at least one of these issues only generically (usually in terms of self-luminous and/or obscuring blobs, and ar- bitrary preferred radii in the accretion disk).

7.1 The sonic point beat-frequency model

Beat-frequency models involve orbital motion at some preferred radius in the disk ( 5.1). A beat-frequency model which uses the magnetospheric radius rM was proposed§ by Alpar & Shaham (1985) to explain the HBO in Z sources ( 6.3; see also Lamb et al. 1985). In the Z sources HBO and kHz QPOs have been§ seen simultaneously, so at least one additional model is required. Miller et al. (1996, 1998a) suggest to continue to use the magnetospheric model for the HBO (see also Psaltis et al. 1999b), and for the kHz QPOs pro- pose the sonic-point beat-frequency model. In this model the preferred radius is the sonic radius rsonic, where the radial inflow velocity becomes supersonic. This radius tends to be near rISCO ( 5.1) but radiative stresses change its location, as required by the observation§ that the kHz QPO frequencies vary. Comparing the HBO and kHz QPO frequencies, clearly r r , so part sonic ≪ M of the accreting matter must remain in near-Keplerian orbits well within rM . At rsonic orbiting clumps form whose matter gradually accretes onto the neutron star following a fixed spiral-shaped trajectory in the frame corotating with their orbital motion (Fig. 16). At the “footpoint”of a clump’s spiral flow the matter hits the surface and emission is enhanced. The footpoint travels around the neutron star at the clump’s orbital angular velocity, so the ob- server sees a hot spot move around the surface with the Keplerian frequency at rsonic. This produces the upper kHz peak at ν2. The high Q of the QPO implies that all clumps are near one precise radius and live for several 0.01 to 0.1 s, and allows for relatively little fluctuations in the spiral flow. The beat frequency at ν1 occurs because a beam of X-rays generated by accretion onto the magnetic poles sweeps around at the neutron star spin frequency νs and hence irradiates the clumps at rsonic once per beat period, which modulates, at the beat frequency, the rate at which the clumps provide matter to their spiral flows and consequently the emission from the footpoints. So, the model predicts ∆ν = ν2 ν1 to be constant at νs, contrary to observations ( 5.5). However, if the clumps’− orbits gradually spiral down, then the observed beat§ frequency will be higher than the actual beat frequency at which beam and clumps interact, because then during the clumps’ lifetime the travel time of matter from clump to surface gradually diminishes. This puts the lower kHz peak closer to the upper one, and thus decrease ∆ν, more so when at higher Lx due to stronger radiation drag the spiralling-down is faster,

31 Figure 16: The clump with its spiral flow, the emission from the flow’s footpoint (dashed lines) and the clump’s interaction with the pulsar beam (lighter shading) in the Miller et al. (1998a) model. as observed (Lamb & Miller 1999, Miller 1999). As the exact way in which this affects the relation between the frequencies is hard to predict, this makes testing the model more difficult. A remaining test is that a number of specific additional frequencies is predicted to arise from the beat-frequency interaction (Miller et al. 1998a), which are different from additional frequencies in, e.g., the relativistic precession model.

7.2 The relativistic precession model Inclined eccentric free-particle orbits around a spinning neutron star show both nodal precession (a wobble of the orbital plane) due to relativistic frame drag- ging (Lense & Thirring 1918), and relativistic periastron precession similar to Mercury’s (Einstein 1915). The relativistic precession model (Stella & Vietri 1998, 1999) identifies ν2 with the frequency of an orbit in the disk and ν1 and the frequency νLF of one of the observed low-frequency (10–100Hz) QPO peaks ( 6.3) with, respectively, periastron precession and nodal precession of this orbit.§ 2 2 2 To lowest order, the relativistic nodal precession is νnod =8π Iν2 νs/c M 32 and the relativistic periastron precession causes the kHz peak separation to 2 1/2 vary as ∆ν = ν2(1 6GM/rc ) , where I is the star’s moment of inertia and r the orbital radius− (Stella et al. 1998, 1999; see also Markovi´c& Lamb 1998). Stellar oblateness affects both precession rates and must be corrected for (Morsink & Stella 1999, Stella et al. 1999b). For acceptable neutron star parameters, there is an approximate match (e.g., Fig. 17) with the observed ν1, ν2 and νLF relations if νLF is twice (or perhaps sometimes four times) the nodal precession frequency, which could in principle arise from a warped disk geometry (Morsink & Stella 1999). In this model the neutron star spin frequencies do not cluster in the 250–350 Hz range ( 5.6) and ∆ν and νburst are not expected to be equal as in beat-frequency interpreta§ tions. A clear prediction is that ∆ν should decrease not only when ν2 increases (as observed) but also when it sufficiently decreases (Fig. 17).

Figure 17: Predicted relations between ν2 and ∆ν (left) and ν1 and νLF as well as ν2 (right) in the relativistic precession model compared with observed values. (Stella & Vietri 1999, Stella et al. 1999b). See §6.3 for a discussion of the data in the right-hand frame.

For a precise match between model and observations, additional free pa- rameters are required. Stella & Vietri (1999) propose that the orbital eccentric- ity e systematically varies with orbital frequency. A critical discussion of the degree to which the precession model and the beat-frequency model can each fit the data can be found in Psaltis et al. (1999b). Vietri & Stella (1998) and Armitage et al. (1999) have performed calculations relevant to the problem of sustaining the tilted orbits required for Lense-Thirring precession in a viscous disk, where the Bardeen-Petterson effect (1975) drives the matter to the or-

33 bital plane. Karas (1999a,b) calculated frequencies and light curves produced by clumps orbiting the neutron star in orbits similar to the ones discussed here. Miller (1999d) calculated the effects of radiation forces on Lense-Thirring pre- cession. Kalogera & Psaltis (1999) explored how the Lense-Thirring precession interpretation constrains neutron star structure. Relatively high neutron star masses (1.8–2M⊙), relatively stiff equations of state, and neutron star spin frequencies in the 300–900Hz range follow from this model. Because it requires no neutron star, but only a relativistic accretion disk to work, the model can also be applied to black holes. Stella et al. (1999b) propose it explains the frequency correlations discussed in 6.3 (Fig. 17). § The idea that the three most prominent frequencies observed are in fact the three main general-relativistic frequencies characterizing a free-particle or- bit is fascinating. However, questions remain. How precessing and eccentric orbits can survive in a disk is not a priori clear (see Psaltis & Norman 1999 for a possible way to obtain these frequencies from a disk). How the flux is mod- ulated at the predicted frequencies, why the basic, orbital, frequency ν2 varies with luminosity, and how the burst oscillations fit in are all open questions. With respect to the burst oscillations the model requires other explanations than neutron star spin ( 4) and these are being explored (Stella 1999, Klu´zniak 1999, Psaltis & Norman§ 1999).

7.3 Photon bubble model A model based on numerical radiation hydrodynamics was proposed by Klein et al. (1996b) for the kHz QPOs in ScoX-1. In this model accretion takes place by way of a magnetic funnel within which accretion is super-Eddington, so that photon bubbles form which rise up by buoyancy and burst at the top in quasi- periodic sequence. In some of the simulations one or two strong QPO peaks are found, whose frequencies increase with accretion rate, as observed (R. Klein, priv. comm.). The model stands out by not requiring rotational phenomena to explain the QPOs, and does not naturally produce beat frequencies. In recent work, attention with respect to this model has shifted to the classical accreting pulsars for which it was originally conceived (Klein et al. 1996a; 6.2). § 7.4 Disk mode models In several models the observed frequencies are identified with oscillation modes of an accretion disk. From an empirical point of view, these models fall into two classes. Some can be seen as “implementations” of a beat-frequency (e.g. Alpar et al 1997) or a precession (Psaltis & Norman 1999) model that they provide ways in accretion disk physics to produce signals at the frequencies

34 occurring in those models. Others produce new frequencies. In all disk models one of the two kHz QPOs is a Keplerian orbital frequency at some radius in the disk. The disk transition layer models (Titarchuk & Muslimov 1997, Titarchuk et al. 1998,1999, Osherovich & Titarchuk 1999a,b, Titarchuk & Osherovich 1999) have evolved into a description (the “two-oscillator model”) where ν1 is identified with the Keplerian frequency at the outer edge of a viscous transition layer between Keplerian disk and neutron-star surface. Oscillations in this layer occur at two low frequencies, producing the noise break and a low frequency QPO ( 6.3). Additionally, blobs described as being thrown out of this layer into a§ magnetosphere oscillate both radially and perpendicular to the disk, producing two harmonics of another low-frequency QPO (in the Z sources this is the HBO) as well as the upper kHz peak. So, altogether this description provides six frequencies which can all fit observed frequencies. Further work on disk oscillations was performed by Lai (1998, 1999), Lai et al. (1999) and Ghosh (1998; see also Moderski & Czerny 1999).

8 Final remark

RXTE has opened up a window that allows us to see down to the very bot- toms of the potential wells of some neutron stars and perhaps to near to the horizons of some black holes. Three millisecond phenomena have been found, whose interpretation relies explicitly on our description of strong-field grav- ity and neutron-star structure. In order to take full advantage of this, it will be necessary to observe the new phenomena with larger instruments (in the 10 m2 class). This will allow to follow the motion of clumps of matter orbit- ing in strong gravity and of hot spots corotating on neutron star surfaces and thereby to to map out curved spacetime near accreting compact objects and measure parameters such as the compactness of neutron stars and the spin of black holes. Acknowledgements: It is a pleasure to acknowledge the help of many colleagues who either made data available before publication, sent originals of figures, read versions of the manuscript or provided insightful discussion: Didier Barret, Lars Bildsten, Deepto Chakrabarty, Wei Cui, Eric Ford, Peter Jonker, Richard Klein, Fred Lamb, Phil Kaaret, Craig Markwardt, Mariano M´endez, Cole Miller, Mike Nowak, Dimitrios Psaltis, Luigi Stella, Tod Stroh- mayer, Rudy Wijnands and Will Zhang. This work was supported in part by the Netherlands Organization for Scientific Research (NWO) and the Nether- lands Research School for Astronomy (NOVA).

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